Question

The reflection of the point $\left(\alpha ,\beta ,\gamma \right)$ in the $xy$-plane is

• A. $\left(\alpha ,\beta ,0\right)$
• B. $\left(0,0,\gamma \right)$
• C. $\left(-\alpha ,-\beta ,\gamma \right)$
• D. $\left(\alpha ,\beta ,-\gamma \right)$

Answer 1

Answer: (D)

Projection of $P\left(\alpha ,\beta ,\gamma \right)xy$-plane is $Q\left(\alpha ,\beta ,0\right)$
If $P^{\prime }\left({\alpha }^{\prime },{\beta }^{\prime },{\gamma }^{\prime }\right)$ is the reflection of $P$ in $xy$-plane, then $Q$ is the mid-point of $P{P}^{\prime }$
$\Rightarrow \left(\alpha ,\beta ,0\right)=\left(\frac{\alpha +{\alpha }^{\prime }}{2},\frac{\beta +{\beta }^{\prime }}{2},\frac{\gamma +{\gamma }^{\prime }}{2}\right)$
$\Rightarrow \frac{\alpha +{\alpha }^{\prime }}{2}=\alpha$, $\frac{\beta +{\beta }^{\prime }}{2}=\beta$, $\frac{\gamma +{\gamma }^{\prime }}{2}=0$
$\Rightarrow {\alpha }^{\prime }=\alpha ,{\beta }^{\prime }=\beta ,{\gamma }^{\prime }=-\gamma$
$\therefore$ Required reflection is $\left(\alpha ,\beta ,-\gamma \right)$